# American Institute of Mathematical Sciences

September  2011, 10(5): 1447-1462. doi: 10.3934/cpaa.2011.10.1447

## Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam

 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042 2 Department of Mathematical Sciences, George Mason University, Fairfax VA, 22030

Received  May 2009 Revised  November 2010 Published  April 2011

In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial directions. The system is modeled using coupled non-linear momentum equations for the axial and transverse displacements. In particular we show that for a class of boundary conditions (beam clamped at the extremes) and uniformly distributed load, there exists a non-uniform equilibrium state. Different models of damping are considered: first, third and fifth order dissipation terms. We show that in all cases in the presence of the damping forces, the excited beam is stable near the equilibrium for any perturbation. An energy estimate approach is used in order to identify the space in which the solution of the perturbed system is stable.
Citation: Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447
##### References:
 [1] A. A. Alqaisia and M. N. Hamdan, Bifurcation and chaos of an immersed cantilever beam in a fluid and carrying an intermediate mass, Journal of Sound and Vibration, 253 (2002), 859-888. doi: 10.1006/jsvi.2001.4072. [2] A. Andrianov and A. Hermans, A VELFP on infinite, finite and shallow water, 17-th International workshop on water waves and floating bodies, Cambridge, UK, aPEIL (2002), 14-17. [3] E. Aulisa, A. Ibragimov, Y. Kaya and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam, Accepted in the "Proceeding of the Seventh AIMS International Conference on Dynamical Systems." [4] E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application, Communications in Computational Physics, 6 (2009), 319-341. doi: 10.4208/cicp.2009.v6.p319. [5] E. Aulisa, S. Manservisi, and P. Seshaiyer, A computational domain decomposition approach for solving coupled flow-structure-thermal interaction problems, Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 13-31. [6] E. Aulisa, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition methodology for solving coupled problems in fluid-structure-thermal interaction, Proceedings of ECCM 2006, Lisbon, Portugal (2006). [7] R. W. Dickey, Dynamic stability of equilibrium states of the extendible beam, Proceedings of the American Mathematical Society, 41 (1973), 94-102. doi: 10.1090/S0002-9939-1973-0328290-8. [8] L. C. Evans, "Partial Differential Equations," AMS, 1998. [9] D. A. Evensen, Nonlinear vibrations of beams with various boundary conditions, AIAA Journal, 6 (1968), 370-372. doi: 10.2514/3.4506. [10] L. Ferguson, E. Aulisa, P. Seshaiyer, Computational modeling of highly flexible membrane wings in micro air vehicles, Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Newport, RI (2006). [11] D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction, Journal of Sound and Vibration, (2007), 231-245. doi: 10.1016/j.jsv.2007.07.032. [12] A. E. Green and J. E. Adkins, "Large Elastic Deformations," Clarendon Press (Oxford), 1970. [13] H. W. Haslach, J. D. Humphrey, Dynamics of biological soft tissue or rubber: Internally pressurized spherical membranes surrounded by a fluid, Int J Nonlin Mech, 39 (2004), 399-420. [14] J. D. Humphrey, "Cardiovascular Solid Mechanics," Springer, 2002. [15] A. I. Ibragimov and P. Koola, The dynamics of wave carpet, P. 2288, OCEAN 2003 MTS/IEEE, proceedings. [16] R. A. Ibrahim, Nonlinear vibrations of suspended cables, Part III: Random excitation and interaction with fluid flow, Applied Mechanics Reviews, 57 (2004), 515-549. doi: 10.1115/1.1804541. [17] J. E. Lagnese, Modelling and stabilization of nonlinear plates, International Series of Numerical Mathematics, 100 (1991), 247-264. [18] C. L. Lou and D. L. Sikarskie, Nonlinear Vibration of beams using a form-function approximation, ASME Journal of Applied Mechanics, 42 (1975), 209-214. doi: 10.1115/1.3423520. [19] C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates, Computers and Structures, 3 (1973), 163-174. [20] J. Padovan, Nonlinear vibrations of general structures, Journal of Sound and Vibration, 72 (1980), 427-441. [21] J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam, Journal of Applied Mathematics, in Press (2009). doi: 10.1155/2009/818269. [22] J. N. Reddy, Finite element modeling of structural vibrations: A review of recent advances, The Shock Vibration Digest, 11, 25-39. [23] J. N. Reddy, An introduction to Nonlinear Finite Element Analysis, Oxford University, 2004. doi: 10.1093/acprof:oso/9780198525295.001.0001. [24] D. L. Russel, A comparison of certain dissipation mechanisms via decoupling and projection techniques, Quart. Appl. Math., XLIX (1991), 373-396. [25] P. Seshaiyer and J. D. Humphrey, A sub-domain inverse finite element characterization of hyperelastic membranes, including soft tissues, ASME J Biomech Engr., 125 (2003), 363-371. doi: 10.1115/1.1574333. [26] W. Shyy, Y. Lian, J. Tang, D. Viieru and H. Liu, Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, 2007. doi: 10.1017/CBO9780511551154. [27] G. Singh, G. V. Rao and N. G. R. Iyengar, Reinvestigation of large amplitude free vibrations of beams using finite elements, Journal of Sound and Vibration, 143 (1990), 351-355. [28] H. Wagner and V. Ramamurti, Beam vibrations-A review, The Shock and Vibration Digest, 9 (1977), 17-24. [29] O. C. Zienkiewicz and R. L. Taylor, "The Finite Element Method," McGraw-Hill, 1993,

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##### References:
 [1] A. A. Alqaisia and M. N. Hamdan, Bifurcation and chaos of an immersed cantilever beam in a fluid and carrying an intermediate mass, Journal of Sound and Vibration, 253 (2002), 859-888. doi: 10.1006/jsvi.2001.4072. [2] A. Andrianov and A. Hermans, A VELFP on infinite, finite and shallow water, 17-th International workshop on water waves and floating bodies, Cambridge, UK, aPEIL (2002), 14-17. [3] E. Aulisa, A. Ibragimov, Y. Kaya and P. Seshaiyer, A stability estimate for fluid structure interaction problem with non-linear beam, Accepted in the "Proceeding of the Seventh AIMS International Conference on Dynamical Systems." [4] E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application, Communications in Computational Physics, 6 (2009), 319-341. doi: 10.4208/cicp.2009.v6.p319. [5] E. Aulisa, S. Manservisi, and P. Seshaiyer, A computational domain decomposition approach for solving coupled flow-structure-thermal interaction problems, Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 13-31. [6] E. Aulisa, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition methodology for solving coupled problems in fluid-structure-thermal interaction, Proceedings of ECCM 2006, Lisbon, Portugal (2006). [7] R. W. Dickey, Dynamic stability of equilibrium states of the extendible beam, Proceedings of the American Mathematical Society, 41 (1973), 94-102. doi: 10.1090/S0002-9939-1973-0328290-8. [8] L. C. Evans, "Partial Differential Equations," AMS, 1998. [9] D. A. Evensen, Nonlinear vibrations of beams with various boundary conditions, AIAA Journal, 6 (1968), 370-372. doi: 10.2514/3.4506. [10] L. Ferguson, E. Aulisa, P. Seshaiyer, Computational modeling of highly flexible membrane wings in micro air vehicles, Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference Newport, RI (2006). [11] D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction, Journal of Sound and Vibration, (2007), 231-245. doi: 10.1016/j.jsv.2007.07.032. [12] A. E. Green and J. E. Adkins, "Large Elastic Deformations," Clarendon Press (Oxford), 1970. [13] H. W. Haslach, J. D. Humphrey, Dynamics of biological soft tissue or rubber: Internally pressurized spherical membranes surrounded by a fluid, Int J Nonlin Mech, 39 (2004), 399-420. [14] J. D. Humphrey, "Cardiovascular Solid Mechanics," Springer, 2002. [15] A. I. Ibragimov and P. Koola, The dynamics of wave carpet, P. 2288, OCEAN 2003 MTS/IEEE, proceedings. [16] R. A. Ibrahim, Nonlinear vibrations of suspended cables, Part III: Random excitation and interaction with fluid flow, Applied Mechanics Reviews, 57 (2004), 515-549. doi: 10.1115/1.1804541. [17] J. E. Lagnese, Modelling and stabilization of nonlinear plates, International Series of Numerical Mathematics, 100 (1991), 247-264. [18] C. L. Lou and D. L. Sikarskie, Nonlinear Vibration of beams using a form-function approximation, ASME Journal of Applied Mechanics, 42 (1975), 209-214. doi: 10.1115/1.3423520. [19] C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates, Computers and Structures, 3 (1973), 163-174. [20] J. Padovan, Nonlinear vibrations of general structures, Journal of Sound and Vibration, 72 (1980), 427-441. [21] J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam, Journal of Applied Mathematics, in Press (2009). doi: 10.1155/2009/818269. [22] J. N. Reddy, Finite element modeling of structural vibrations: A review of recent advances, The Shock Vibration Digest, 11, 25-39. [23] J. N. Reddy, An introduction to Nonlinear Finite Element Analysis, Oxford University, 2004. doi: 10.1093/acprof:oso/9780198525295.001.0001. [24] D. L. Russel, A comparison of certain dissipation mechanisms via decoupling and projection techniques, Quart. Appl. Math., XLIX (1991), 373-396. [25] P. Seshaiyer and J. D. Humphrey, A sub-domain inverse finite element characterization of hyperelastic membranes, including soft tissues, ASME J Biomech Engr., 125 (2003), 363-371. doi: 10.1115/1.1574333. [26] W. Shyy, Y. Lian, J. Tang, D. Viieru and H. Liu, Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, 2007. doi: 10.1017/CBO9780511551154. [27] G. Singh, G. V. Rao and N. G. R. Iyengar, Reinvestigation of large amplitude free vibrations of beams using finite elements, Journal of Sound and Vibration, 143 (1990), 351-355. [28] H. Wagner and V. Ramamurti, Beam vibrations-A review, The Shock and Vibration Digest, 9 (1977), 17-24. [29] O. C. Zienkiewicz and R. L. Taylor, "The Finite Element Method," McGraw-Hill, 1993,
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